Let's break down the problem systematically.

(a) Finding the equation for the Production Possibility Frontier (PPF)

Vivian has 100 units of labor, and labor can be allocated between producing good $x$ and good $y$. Let $L_x$ and $L_y$ be the amounts of labor allocated to producing goods $x$ and $y$, respectively. Therefore, we have the total labor constraint:

$L_x + L_y = 100$

The production functions for goods $x$ and $y$ are given as:

$x = 0.5 \sqrt{L_x} \quad \text{and} \quad y = \sqrt{L_y}$

We want to derive the PPF, which describes the trade-off between producing $x$ and $y$. To do this, solve both production functions in terms of $L_x$ and $L_y$:

1. From $x = 0.5 \sqrt{L_x}$, solve for $L_x$: $L_x = \left( \frac{x}{0.5} \right)^2 = 4x^2$

2. From $y = \sqrt{L_y}$, solve for $L_y$: $L_y = y^2$

Substitute $L_x = 4x^2$ and $L_y = y^2$ into the total labor constraint $L_x + L_y = 100$:

$4x^2 + y^2 = 100$

This is the equation of the Production Possibility Frontier (PPF). It describes the feasible combinations of goods $x$ and $y$ that can be produced with the available 100 units of labor.

(b) Finding the optimal pair $(x, y)$

Vivian’s utility function is given as:

$u(x, y) = x + 0.2y$

We need to maximize this utility function subject to the PPF constraint $4x^2 + y^2 = 100$. To solve this, we will use the Lagrange multiplier method.

The Lagrangian is:

$\mathcal{L}(x, y, \lambda) = x + 0.2y + \lambda (100 - 4x^2 - y^2)$

Take partial derivatives with respect to $x$, $y$, and $\lambda$:

1. $\frac{\partial \mathcal{L}}{\partial x} = 1 - 8x\lambda = 0 \quad \Rightarrow \quad \lambda = \frac{1}{8x}$
2. $\frac{\partial \mathcal{L}}{\partial y} = 0.2 - 2y\lambda = 0 \quad \Rightarrow \quad \lambda = \frac{0.1}{y}$
3. $\frac{\partial \mathcal{L}}{\partial \lambda} = 100 - 4x^2 - y^2 = 0 \quad \Rightarrow \quad 4x^2 + y^2 = 100$

From equations (1) and (2), equate the expressions for $\lambda$:

$\frac{1}{8x} = \frac{0.1}{y}$

Solve for $y$:

$y = 0.8x$

Substitute $y = 0.8x$ into the PPF equation $4x^2 + y^2 = 100$:

$4x^2 + (0.8x)^2 = 100$ $4x^2 + 0.64x^2 = 100$ $4.64x^2 = 100 \quad \Rightarrow \quad x^2 = \frac{100}{4.64} \quad \Rightarrow \quad x^2 \approx 21.55$ $x \approx \sqrt{21.55} \approx 4.64$

Now, use $y = 0.8x$ to find $y$:

$y \approx 0.8 \times 4.64 \approx 3.71$

Conclusion:

The optimal pair $(x, y)$ that maximizes Vivian’s utility is approximately:

$(x^*, y^*) \approx (4.64, 3.71)$

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Would you like further details on any of the steps, or do you have any specific questions?

Here are five questions to expand on this topic:

1. How does the shape of the PPF change if the production functions change?
2. How would the optimal pair $(x, y)$ change if Vivian's utility function had different weights on $x$ and $y$?
3. What is the economic interpretation of the Lagrange multiplier $\lambda$ in this context?
4. How would a change in the total labor supply (e.g., from 100 to 120) affect the PPF and the optimal solution?
5. What happens if there is an external trade opportunity, allowing Vivian to exchange $x$ and $y$ with the outside world?

Tip: In maximization problems, the Lagrange multiplier $\lambda$ often represents the marginal utility of relaxing the constraint by one unit.